Table Of ContentA KINETIC SELECTION PRINCIPLE FOR CURL-FREE VECTOR FIELDS OF
UNIT NORM
PIERREBOCHARD† ANDPAULPEGON‡
7
1
0 Abstract. This articleis devoted to the generalization ofresults obtained in2002 in[8] byJabin,
2 OttoandPerthame. Intheirarticletheyproved thatplanar vector fields takingvalue intotheunit
sphere of the euclidean norm and satisfying a given kinetic equation are locally Lipschitz. Here,
n
we study the same question replacing the unit sphere of the euclidean norm by the unit sphere of
a
any norm. Under natural asumptions on the norm,namely smoothness and aqualitative convexity
J
property,thatistobeofpowertypep,weprovethatplanarvector fieldstakingvalueintotheunit
1 1
sphere of such a norm and satisfying a certain kinetic equation are locally -Hölder continuous.
1 p−1
Furthermore we completely describe the behaviour of such a vector field around singular points as
avortex associated tothe norm. As ourkineticequation impliesforthevector fieldtobecurl-free,
]
P thiscanbeseenasaselectionprincipleforcurl-freevector fieldsvaluedinspheres ofgeneral norms
whichrulesoutline-likesingularities.
A
.
h
t
a
m Contents
[
Preliminaries and notations 1
1 1. Introduction 2
v 2. Modulus of convexity and vortices 5
1
2.1. Properties of the normal field 6
3
2.2. Modulus of convexity 7
9
2 3. Averaging formula 11
0 4. Kinetic formulation 13
. 4.1. Kinetic formulation and curl-free vector fields of norm one 13
1
0 4.2. Direction conservation and trace theorem 14
7 4.3. The regularity theorem 21
1 References 25
:
v
i
X
r Preliminaries and notations
a
Our notations in this article will be standard. Beware that we will consider in this article maps
insteadofequivalence classesfor the almosteverywhererelation; allmaps willbe takenBorelmeasur-
able. More specifically, let Ω,E R2. We will denote by 1 (Ω,E) the set of Borel maps m which
⊂ Lloc
are locally integrable and such that m(x) B for all x in Ω. Of course we do not lose generality with
∈
†Université Lyon 1,CNRS UMR5208,Institut CamilleJordan,69622Villeurbanne,France
‡Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay
Cedex,France
E-mail addresses: [email protected], [email protected].
1
2 REGULARITY OF CURL-FREE VECTOR FIELDS
this choice. For x R2, x⊥ will denote its rotation by an angle +π/2. The euclidean norm will be
∈
denoted by and a general norm by .
|·| k·k
1. Introduction
The goal of this paper is the generalization of regularity results on solutions of a kinetic equation
obtainedbyJabin,OttoandPerthameintheirarticle[8]. Inorderforthe readertounderstandwhere
wearestartingfrom,wewillbeginbybrieflyrecallingtheirresults. LetΩ R2 beanopenset. Jabin,
Otto and Perthame were interested in the study of the following micromag⊂netic energy on H1(Ω,R2):
1 1
(1.1) E (m)=ε Dm2+ (1 m2)2+ −1curlm2,
ε ˆΩ| | ε ˆΩ −| | ε ˆR2|∇ |
where m is extended by zero outside of Ω. They associate to the measurable vector field m: Ω R2
→
the quantity:
χ : Ω S1 0,1
m
(x×,s) −→ 1 { } .
{m(x)·s>0}
7−→
The introduction of such a quantity is motivated by the following averaging formula: let u S1, then
∈
1
(1.2) u= 1 sd 1(s)
2ˆS1 {u·s>0} H
Anobviousconsequenceof (1.2)isthatifwesupposethat m =1almosteverywhere,thenforalmost
| |
every x in Ω,
1
m(x)= χ (x,s)sd 1(s).
2ˆS1 m H
They proved then the following theorem:
Theorem 1.1 (Jabin, Otto, Perthame). Let m be a zero-state energy, that is a limit of a sequence
(m ) H1(Ω,R2) with E (m ) 0. Then m = 1 almost everywhere and m satisfies the
ε ε>0 ε ε
⊂ −ε−→−→0 | |
kinetic equation:
(1.3) s S1, s⊥ χ (,s)=0 in ′(Ω).
m
∀ ∈ ·∇ · D
Classical results on the regularity of the average of solutions of kinetic equations tell us (see [6]
for example) that equation (1.3) implies for m to be in H12(Ω,R2). But in the case of this particular
kinetic equation, it turns out that combining (1.3) with m = 1 implies a much stronger regularity.
| |
More precisely, the following holds:
Theorem 1.2 (Jabin, Otto, Perthame). Let m be a measurable vector field on Ω satisfying m = 1
| |
almost everywhere and equation (1.3). Then m is locally Lipschitz continuous inside Ω except at a
locally finite set of singular points. Furthermore, for every singular point p, there is α 1,1 such
∈{− }
that in any convex neighborhood of p,
x p
m(x)=α − a.e.
x p
| − |
Before we state our own results, let us give a bit more insight into the two previous theorems.
Looking at Theorem 1.1, we expect a zero-state energy m to be of modulus one and curl-free because
of the energy (1.1). As a consequence, the curl-free information about m has to be encompassed
into equation (1.3). It is indeed easy to see that if a modulus one vector field checks equation (1.3)
then it is curl-free; we will prove this in our more general setting in Proposition 4.1. Keeping this in
REGULARITY OF CURL-FREE VECTOR FIELDS 3
mind, we can know look at Theorem 1.2 in the following way: equation (1.3) is a selection principle
amongcurl-freevectorfields takingvaluesinto S1 which, roughlyspeaking,doesnotallowforline-like
singularities.
The question we are addressing here is the following: can we find a kinetic equation which will act
as a selection principle for curl-free vector field taking value into the unit sphere of any norm instead
of S1? The purpose of this article is to prove that if we make simple geometric assumptions on our
norm, the answer to this question is yes. In the spirit of Theorem 1.2, we will then prove that this
kinetic equation implies some regularity on the vector field itself.
Our paperwill be dividedinto three parts. The firstone willbe devotedto prerequisitesongeneral
norms and their modulus of convexity, the second one to the study of a generalized avering formula
in the spirit of (1.2) and the last one to our main result, i.e. the regularity of solutions to the kinetic
equation we are going to introduce.
Before stating our results, we need some definitions on norms and on the way to measure their
convexity.
Definition 1.1. Let beanormandB itsclosedunitball. Wecallmodulus of convexity associated
k·k
to B the quantity ω : [0,2] [0,1] defined by:
B
→
x+y
(1.4) ω (δ)=inf 1 : x = y =1 and x y δ .
B (cid:26) −(cid:13) 2 (cid:13) k k k k k − k≥ (cid:27)
(cid:13) (cid:13)
(cid:13) (cid:13)
Definition 1.2. Let p 0. We say t(cid:13)hat a n(cid:13)orm is of power type p if there is a constant K >0 such
≥
that:
(1.5) δ [0,2], Kδp ω (δ)
B
∀ ∈ ≤
As an example, one should keep in mind that the euclidean norm is of power type 2 and that
more generally, for p > 1, lp norms are of power type max p,2 : this is a consequence of Clarkson’s
{ }
inequalities and can be found in [10]. The main reason for introducing theses two notions is that
we need some property on the norm in order to obtain a kinetic equation which acts as a selection
principle excluding line-like singularities. It turns out that the power type p property is exactly the
goodpropertyto lookat. We referthe readerto the beginning ofourthirdsectionfor amoredetailed
discussion.
The first important result of our article is an avering formula in the spirit of (1.2):
Theorem 1.3. Let B R2 be a symmetric convex body, that is the unit ball of some norm, then:
⊂
1
(1.6) x ∂B, x= 1 n (s)d 1(s),
∀ ∈ 2ˆ {x·s>0} B⊥ H
∂B⊥
where n (s) stands for the unit normal to ∂B⊥ at s.
B⊥
Remark 1.1. Note that formula (4.1) generalizes easily in any dimension in the case of the euclidean
sphere. We refer the reader to [1] for such a formula and its use to prove a rigidity result obtained in
the study of an analog of equation (1.3) in dimension greater than 2. It would be nice to know if we
can obtain an averaging formula like (1.6) in greater dimension for a general symmetric convex body.
The following corollary of Theorem 1.3, which is crucial to prove our main result, is immediate.
Corollary 1.4. Let . be a norm, B its closed unit ball and m: Ω R2 a Borel vector field such
kk →
that m =1. We associate to m the quantity:
k k
χ : Ω ∂B⊥ 0,1
m
(×x,s) −→ 1 { } .
{m(x)·s>0}
7−→
4 REGULARITY OF CURL-FREE VECTOR FIELDS
Then, for all x in Ω,
1
(1.7) m(x)= χ (x,s)n (s)d 1(s).
2ˆ m B⊥ H
∂B⊥
We need a last definition to state our main result:
Definition 1.3. Let . be a norm with unit ball B and its dual norm defined as usual by the
kk k·k⋆
equality:
x = sup x,y .
k k⋆ h i
kyk≤1
We call vortex associated to the norm the function V : x at the points where this
quantity makes sense. The map x .k·kbeing convex, VBis we7→ll-d∇efixnke·kd⋆almost everywhere on R2
B
7→ kk
according to Rademacher’s theorem.
We can now state the main theorem of our article.
Theorem 1.5. Let . be a norm of power type p and B its closed unit ball such that ∂B is a C1
submanifold in R2. Lketkm: Ω R2 be a Borel vector field such that m =1 and satisfying:
→ k k
(1.8) s ∂B⊥, n (s)⊥ χ (,s)=0 in ′(Ω).
∀ ∈ B⊥ ·∇ m · D
Then m is locally 1 -Hölder continuous outside a locally finite set of singular point. Furthermore,
p−1
for every singular point p, there is α 1,1 such that in any convex neighborhood of p,
∈{− }
m(x)=αV (x p) almost everywhere,
B
−
where V is the vortex associated to . defined in Definition 1.3.
B
kk
Remark 1.2. Noticethatthetwohypothesesonthenormarenotredundant: wecanfindnormswhich
are not of power type p for any p and whose unit sphere is a C1 submanifold. Take for example the
norm associated to the l1 unit ball whose angles have been smoothed out. On the other hand, an
example ofa type p normwhose unit ballis not a C1 submanifoldis givenby x = x + x with
k k k k1 k kp
p>2. The reader interested in the links between convexity and smoothness of the ball of a norm will
consult with profit Section 1.e. of [10].
In order to see how Theorem 1.5 could be used, let us note that Theorem 1.2 itself has interesting
consequences. For example,together with clever commutatorestimates, De Lellis and Ignatprovedin
[2] the following:
Theorem 1.6 (De Lellis, Ignat). Let Ω be an open set in R2 and r [1,3]. Then every m in
∈
Wr1,r(Ω,S1):= m Wr1,r(Ω,R2):curlm=0 in ′(Ω) and m =1 a.e.
curl n ∈ D | | o
is locally Lipschitz continuous inside Ω except at a locally finite set of singular points, where
W1r,r(Ω,R2) stands for the usual fractional Sobolev space.
A first natural question would be to ask if we can obtain such a theorem when S1 is changed into
theunitsphereofanormsatisfyingthe hypothesesofTheorem1.5. Anothernaturalquestion,inview
of Theorem 1.1, would be to ask if we can obtain an analog of Theorem 1.1 if we change the energy
(1.1) into:
1 1
E˜ (m)=ε Dm2+ (1 m 2)2+ −1curlm2,
ε ˆΩ| | ε ˆΩ −k k ε ˆR2|∇ |
REGULARITY OF CURL-FREE VECTOR FIELDS 5
where . is now a norm satisfying the hypotheses of Theorem 1.5. Note that the proof of both
kk
Theorem1.6 andTheorem1.1 rely on the notion of entropy, that is, in this context, smoothfunctions
Φ: S1 R2 such that for all open Ω in R2,
→
m C∞ (Ω,S1), [Φ m]=0.
∀ ∈ curl ∇· ◦
Wereferthereaderto[3]and[7]formoredetailsandapplicationsofthisnotion. Thereforeafirststep
in order to obtainanalogs of Theorems 1.1 and 1.6 would be to developan adapted notion of entropy.
We will adress this question in a paper to come.
2. Modulus of convexity and vortices
Our main sources for this section are [9] and [10] on the geometry of norms and [4] on convex
analysis. For the reader’s comfort, let us collect some general facts on convex functions and convex
bodies:
A function f :Rd R + is said convex if f((1 t)x+ty) (1 t)f(x)+tf(y) for all
• x,y Rd, t [0,1]→. Its∪do{ma∞in}is defined as Dom(f):−= x:f(x)≤< −.
∈ ∈ { ∞}
The subdifferential of f at x is defined as ∂f(x) := p : y,f(y) f(x)+ p,y x . It is
• ◦ { ∀ ≥ h − i}
nonempty if x Dom(f). In that case, f is differentiable at x if and only if its subdifferential
∈
is a singleton, in which case ∂f(x)= f(x) .
{∇ }
The Legendre-Fenchel transform of f is defined as
•
f⋆(p)=supx p f(x).
· −
x
If f is a proper ( + ) convex lsc function, so is f⋆, and f =f⋆⋆.
6≡ ∞
In that case, ∂f⋆ is the inverse of ∂f in the sense that
•
(2.1) p ∂f(x) x ∂f⋆(p).
∈ ⇐⇒ ∈
If B is a convex body and its associated norm, then χ⋆ = .
• k·k B k·k⋆
If x ∂B, we say that u is a normal vector to B at x if
• ∈
y B, u (y x) 0.
∀ ∈ · − ≤
Notice that the set of normal vectors, denoted by (x), is a convex cone which is precisely
B
N
∂χ (x).
B
If x ∂B, the following relation between the subdifferentials of and χ holds:
B
• ∈ k·k
(2.2) ∂ =∂χ (x) B⋆, or equivalently ∂ = (x) B⋆.
x B x B
k·k ∩ k·k N ∩
There is a unique unit normal vector at x ∂B if and only if the associated norm is differ-
• ∈
entiable at x. In general this is true for 1-almost every x ∂B, and it is the case for all
H ∈
x ∂B for instance if ∂B is a 1 submanifold.
∈ C
Beforegetting to the heartofthe matter,letus explainwhythe normshouldsatisfysome property
forthekineticequationtoactasaselectionprinciplerulingoutline-likesingularities. Forthatpurpose,
letustakeanotherlookatthevortexV associatedtoanorm ofunitballBasdefinedinDefinition
B
1.3. It is clear that V is curl-free in ′(R2). Indeed, taking ϕk·k C∞(R2),
B D ∈ 0
∂ ϕV ∂ ϕV dx= ϕ⊥ x dx
ˆR2 1 B,2− 2 B,1 ˆR2∇ ·∇k k⋆
= ϕ⊥ x dx=0,
ˆR2∇·(cid:2)∇ (cid:3)k k⋆
6 REGULARITY OF CURL-FREE VECTOR FIELDS
where we used the fact that V is the distributional gradient of by Rademacher’s theorem. Fur-
B
k·k
thermore, V takes its values into ∂B. Indeed by (2.1), for almost every x (those where is
B k·k⋆
differentiable, i.e. where V is defined), one has p = V (x) x ∂ , so that p is a point
B B p
⇐⇒ ∈ k·k
where the subdifferential ∂ contains a nonzero vector, i.e. a point on the boundary of the ball.
p
k·k
Consequently V (x) = p ∂B. We are going to prove that V is always a solution to the kinetic
B B
∈
equation (1.8). As a consequence, if we want this kinetic equation to rule out line-like singularities
then V should not exhibit such singularities, and the regularity of V is related to the convexity of
B B
B, as we will see in Proposition 2.5. This is the fundamental reason why we need some convexity
property on our norm in order to obtain Theorem 1.5.
2.1. Properties of the normal field. We start off by stating some useful properties of the normal
field
n : ∂B (S1)
B
x −→ P(x) S1,
B
7−→ N ∩
where (x) is the normal cone to B at x.
B
N
Lemma 2.1. Let B be the unit ball of a norm and n the application defined above. Then, for
B
1-almost every y in S1, there is a unique x in ∂B such that y n (x).
B
H ∈
Proof. Let us consider:
n˜ : ∂B (∂B⋆) n˜ : ∂B⋆ (∂B)
B B⋆
−→ P −→ P
x (x) ∂B⋆, x (x) ∂B.
B B⋆
7−→ N ∩ 7−→ N ∩
By (2.1) and (2.2), it is easy to see that p n˜ (x) if and only if x n˜ (p). Since ∂B⋆ admits a
B B⋆
∈ ∈
normalvectorfor 1-almosteveryp,wegetthatthereisauniquexsatisfyingp n˜ (x)for 1-almost
B
every p in ∂B⋆. WHe get uniqueness for n by using the fact that the projectio∈n from S1 toH∂B⋆ is a
B
bi-Lipschitz bijection. (cid:3)
Remark 2.1. In the rest of the paper, if y S1 is such that there is a unique x ∂B satisfying
y n (x), we will abusively denote it by x =∈n−1(y). One can deduce from the pre∈vious proof that
th∈is inBin particular the case if ∂B⋆ is a C1 submBanifold of R2.
Proposition 2.2. Let be a norm of unit ball B, then for almost every x,
k·k
x
V (x)=n−1 .
B B (cid:18) x (cid:19)
| |
Proof. Weknowthat istheLegendre-Fencheltransformofχ . Consequently isdifferentiable
k·k⋆ B k·k⋆
at x if and only if x is in the normal cone of a unique p ∂B, that is if and only if n−1 x is well
∈ B (cid:16)|x|(cid:17)
.
defined, in which case n−1 x =p= =V (x). By the previous lemma, it is true for almost
every x. B (cid:16)|x|(cid:17) ∇xk·k⋆ B (cid:3)
We can now prove as promised that V is alwayssolution to the kinetic equation (1.8). Indeed, let
B
Ω R2, ϕ C∞(Ω), and s ∂B. Using Proposition 2.2, for almost every x Ω, V (x)=n−1 x ,
⊂ ∈ 0 ∈ ∈ B B (cid:16)|x|(cid:17)
and
ϕ n (s)⊥χ (x,s)dx= ϕ n (s)⊥dx.
ˆΩ∇ · B⊥ VB ˆΩ∩{n−1 x ·s>0}∇ · B⊥
B |x|
(cid:0) (cid:1)
REGULARITY OF CURL-FREE VECTOR FIELDS 7
But for almost every s, n−1 x s>0 is an half-space whose boundary is the line L passing
n B (cid:16)|x|(cid:17)· o
through 0 and directed by n (s⊥), whose unit normal is n (s⊥)⊥ = n (s). As a consequence,
B ± B ∓ B⊥
using Stokes formula and the fact that ϕ is supported in Ω,
ϕ n (s)⊥dx= n (s)⊥ n (s) ϕdx=0.
ˆΩ∩{n−1 x ·s>0}∇ · B⊥ ± B⊥ · B⊥ ˆΩ∩L
B |x|
Let us define a last qua(cid:0)nti(cid:1)ty that we will need later. For u,v R2 which are not colinear, we define
∈
the convex cone C(u,v) generated by u,v as
C(u,v):= λ u+λ v :(λ ,λ ) R+ R+ .
1 2 1 2
∈ ×
(cid:8) (cid:9)
Lemma 2.3. Let be a strictly convex and differentiable (outside 0) norm of closed unit ball B.
Then n :∂B S1k·iks monotonic in the sense that
B
→
w C(u,v) n (w) C(n (u),n (v)).
B B B
∈ ⇐⇒ ∈
Proof. Notice that C(u,v)is (by Hahn-BanachTheorem)the intersectionof the half-spaceswhichare
delimited by vector hyperplanes and which contain u and v. First, let us show that, given any such
half-space H, n is a bijection between ∂B H and a half-circle of S1. Since n is continuous and
B B
∂B H is compact and connected, n (∂B ∩H) is a compact and connected subset of S1, hence it
B
is a∩closed arc of S1. Since ∂B H contain∩s two symmetric points x ∂B, so does n (∂B H),
B
∩ ± ∈ ∩
hence it contains a half-circle [θ,θ+π]. It may not be larger than that, otherwise it would contain a
nontrivialarc and its symmetric,hence ∂B H as well since n is homeomorphicand antisymmetric.
B
∩
This cannot be true. This proves that n sends half-spaces of ∂B to half-circles and it is clear that it
B
realizes a bijection between the set of half-spaces of ∂B and the set of half-circles of S1.
Now w C(u,v) means that w belongs to any half-space (delimited by a vector hyperplane)
∈
containing u,v, which means that n (w) belongs to any half-circle containing n (u),n (v) by what
B B B
we just proved. This exactly means that n (w) C(n (u),n (v)). (cid:3)
B B B
∈
2.2. Modulus of convexity. In this section we give an alternate definition of the modulus of con-
vexity, which offers some advantages: an intuitive interpretation is given in Proposition 2.4, and a
characterizationinterms ofthe regularityofthe vortexisgiveninProposition2.5. This will be useful
to prove the sharpness of our main theorem. Of course, it is essentially equivalent to Definition 1.1
which is given in the introduction, as showed in Remark 2.3.
Definition 2.1. We define the modulus of convexity of B as the greatest nondecreasing function
ρ :[0,2] R such that for all x,y ∂B and all u (x),
B + B
→ ∈ ∈N
(2.3) u y u x u ρ ( y x ).
· ≤ · −k k⋆ B k − k
We say that B is of power type p 2 if ρ (δ) Cδp for some C > 0 and that it is elliptic if it is of
B
≥ ≥
power type 2.
Remark 2.2. Notice that
y ∂B, u y u x u ρ ( y x )
∀ ∈ · ≤ · −k k⋆ B k − k
if and only if
y Rd, χ (y) χ (x)+u (y x)+ u ρ ( y x ).
∀ ∈ B ≥ B · − k k⋆ B k − k
This last inequality is reminiscent of the inequality f(y) f(x)+p (y x) for p ∂f(x) which
≥ · − ∈
holdsforarbitraryconvexfunctions. Theextraterm u ρ ( y x )measureshowmuchB isconvex
k k⋆ B k − k
around its boundary.
8 REGULARITY OF CURL-FREE VECTOR FIELDS
Remark 2.3. The two definitions of modulus of convexity are equivalent in the sense that:
ρ (δ/2) ω (δ) ρ (δ)/2.
B B B
≤ ≤
Proof. The proof is quite easy. Take x,y ∂B and u ((x+y)/2) such that u = 1. Notice
∈ ∈ NB k k⋆
that u ∂ (because u =1), hence by convexity:
∈ x+2yk·k k k⋆
x+y x y
1= x +u − .
k k≥(cid:13) 2 (cid:13) · 2
(cid:13) (cid:13)
By definition of ρ one has: (cid:13) (cid:13)
B (cid:13) (cid:13)
x+y x+y
u y u ρ y ,
· ≤ · 2 − B(cid:18)(cid:13) − 2 (cid:13)(cid:19)
(cid:13) (cid:13)
or equivalently (cid:13) (cid:13)
(cid:13) (cid:13)
y x y x
u − ρ k − k .
B
· 2 ≤− (cid:18) 2 (cid:19)
Putting this in the previous inequality yields:
x+y x y y x
1 u − 1 ρ k − k .
(cid:13) 2 (cid:13)≤ − · 2 ≤ − B(cid:18) 2 (cid:19)
(cid:13) (cid:13)
Since ωB is precisely the lar(cid:13)(cid:13)gest m(cid:13)(cid:13)ap for which the inequality x+2y ≤1−ωB(ky−xk) holds for all
x,y ∂B, it follows that: (cid:13) (cid:13)
∈ (cid:13) (cid:13)
ρ (δ/2) ω (δ)
B B
≤
for all δ [0,2].
∈
Now let us prove the other inequality. This time we take x,y ∂B, u (x) with u = 1. By
∈ ∈ N k k⋆
definition we know that:
x+y
1 ω ( y x ),
(cid:13) 2 (cid:13)≤ − B k − k
(cid:13) (cid:13)
but since u =1, one has: (cid:13) (cid:13)
k k⋆ (cid:13) (cid:13)
x+y x+y y x y x
u =u x+u − =1+u − .
(cid:13) 2 (cid:13)≥ · 2 · · 2 · 2
(cid:13) (cid:13)
Putting these two toget(cid:13)her, on(cid:13)e gets:
(cid:13) (cid:13)
u y u x 2ω ( y x ),
B
· ≤ · − k − k
and by homogenity if u has arbitrary norm:
u y u x u 2ω ( y x ).
· ≤ · −k k⋆· B k − k
Since ρ is the largestfunction for which such an inequality holds for all x,y and u (x), one has
B B
∈N
2ω (δ) ρ (δ)
B B
≤
which concludes the proof. (cid:3)
As indicated by its name, the greater ρ is, the more convex B is. For example, taking a look at
B
Figure 1, it is easy to see that the modulus of convexityassociatedto the l1 normis constantequal to
0 on a neighborhood of 0. On the other side, the euclidean ball is the most convex of all in the sense
that if ω is its modulus of convexity, for all symmetric convex body B,
2
(2.4) δ [0,2], ω (δ) ω (δ).
B 2
∀ ∈ ≤
We refer the reader to [11] for a proof of this inequality.
REGULARITY OF CURL-FREE VECTOR FIELDS 9
Figure 1. Unit ball of . and vortex associated to . , singular along the axes.
1 ∞
kk kk
Remark 2.4. Using the symmetry of S1, ρ can easily be computed and one finds:
2
δ2
ρ (δ)=1 1 δ2 .
2
−p − δ→∼0 2
As a consequence of this equality, of inequality (2.4) and Remark 2.3, there is no norm of power type
p with p<2.
Throughouttherestofthispaper,wewillusethemodulusofconvexityρ asdefinedinthissection.
B
Thefollowingpropositiongivesanintuitivecharacterizationofconvexbodiesofpowertypep. Itstates
that they are convex bodies whose boundary is locally the graph of a map which is above that of the
map t tp around 0.
7→| |
Proposition 2.4. Let x ∂B, u a normal vector of unit euclidean norm. Take γ :] ε,+ε[ ∂B
∈ − →
a local parameterization of ∂B around x = γ(0), with direct orientation and unit speed. We denote
by τ = γ˙(0) the unit tangent at x and ν = τ⊥ the inner normal. In the basis (τ,ν), we write
γ(t)=x+a(t)τ +b(t)ν. Then B is of type p if and only if
(2.5) b(t) C a(t)p.
≥ | |
If B is of class C2 and κ(x) denotes the curvature at x, B is elliptic if and only if for some C >0:
x ∂B, κ(x) C.
∀ ∈ ≥
Proof. Writing inequality (2.3) with y =γ(t) one gets
.
b(t)=u (γ(t) x) C γ(t) xp C a(t)p
− · − ≤− | − | ≤− | |
which yields (2.5). Conversely,since b=o(a) the inequality b(t) C a(t)p implies (2.3) for x close to
≥ | |
y, which is enough to get it for all x,y.
If ∂B is of class C2 and so is γ, by definition one has
γ(t)=x+tγ˙(0)+t2/2 γ¨(0)+o(t2)=x+tτ +κ(x)t2/2 ν+o(t2).
· ·
Consequently a(t)=t+o(t2) and b(t)=κt2/2+o(t2), thus B being elliptic, one has:
C(t+o(t2))2 κt2/2+o(t2).
≤
Dividingbyt2andt 0yieldsκ 2C and∂Bhaspositivecurvature. Theconverseisstraightforward.
→ ≥ (cid:3)
The next proposition establishes a link between the power type p property of a norm and the
smoothness of its vortex.
10 REGULARITY OF CURL-FREE VECTOR FIELDS
Proposition 2.5. Let B be a convex body associated to the norm . Then B is of power type p if
. k·k
and only if is differentiable everywhere outside 0 and V = is 1 -Hölder continuous far
k·k⋆ B ∇k·k⋆ p−1
from 0, in the sense that for some constant H >0 one has:
H 1
V (v) V (u) u v p−1,
k B − B k≤ δp−11k − k⋆
for all u , v δ and all δ >0.
k k k k≥
Proof. Take x,y ∂B and u (x),v (y) with u,v nonzero. One has
B B
∈ ∈N ∈N
u y u x u ρ ( y x ),
· ≤ · −k k⋆ B k − k
v x v y v ρ ( y x ),
· ≤ · −k k⋆ B k − k
and by summing these one gets:
2min( u , v )ρ ( y x ) (v u) (y x).
k k⋆ k k⋆ B k − k ≤ − · −
Now if B is of power type p then
(v u) (y x) v u y x
C y x p ρ ( y x ) − · − k − k⋆k − k,
B
k − k ≤ k − k ≤ min( u , v ) ≤ min( u , v )
k k⋆ k k⋆ k k⋆ k k⋆
thus
1
u v p−1
y x k − k⋆ ,
k − k≤(cid:18)Cmin( u , v )(cid:19)
k k⋆ k k⋆
which yields
H 1
y x u v p−1.
k − k≤ δp−11k − k⋆
provided that u , v δ, for some constant H depending only on the norm.
k k⋆ k k⋆ ≥
Conversely, take x ∂B, u (x) with u =1. We want to show that for all y B:
∈ ∈NB k k⋆ ∈
u y u x C y xp.
· ≤ · − | − |
Takex aminimizer ofg(y)=χ (y)+(u (x y) C|y−x|p). We wanttoshow thatforsmallenough
0 B · − − p
C, g(x ) 0. The necessary optimality condition reads
0
≥
u :=u+C(x x)p−1 ∂χ (x ).
0 0 B 0
− ∈
Necessarily, x B thus
0
∈ u u δC x x p−1 δC2p−1
k 0− k⋆ ≤ k 0− k ≤
where δ := sup v . If C is small enough, say Cδ2p−1 1/2 then both u , u 1/2. On
kvk=1k k⋆ ≤ k k⋆ k 0k⋆ ≥
the other hand one has by hypothesis:
x x p−1 2Hp−1 u u 2δCHp−1 x x p−1.
k 0− k ≤ k 0− k⋆ ≤ k 0− k
Now if one takes C even smaller, so that 2δCHp−1 < 1, one must have x = x and u = u , which
0 0
implies that minR2g =g(x0)=g(x)=0 and the following holds:
y B, u y u x C y xp/p with C =min(2−p,1/(2δHp−1)).
∀ ∈ · ≤ · − | − |
If u has arbitrary norm, this becomes:
y B, u y u x C′ u y x p
∀ ∈ · ≤ · − k k⋆k − k
where C′ =C/p. (cid:3)
As a corollary,one gets the following regularity theorem.
